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Plot the function to obtain a qualitative view of its \
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\>", "Text"],

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Cell[TextData[StyleBox["Review of Maxima & Minima of a Function of Two \
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We can use the graph of a function of two variables, viewed either \
as a surface in space or as a contour map, to check conclusions about the \
location and classification of critical points.  This provides a visual aid \
in determining maxima, minima and saddle points.\
\>", "Text",
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Read through the explanation of how to find critical points and \
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Cell[TextData[{
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Now set the two partial derivatives = 0 and solve the system.\
\>", \
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Cell[TextData[{
  "We get no information about these critical points,  (0, 1) and (0,-1).  We \
need some other means of figuring out what is happening at these points. \
First, let's draw the graph of ",
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The plot in three dimensions is hard to interpret from the standard \
viewpoint.  We might try another viewpoint, or instead we might try a contour \
plot and look at the level curves.  Remember that the contour plot shades \
darkest where the graph is lowest and lightest where the graph is highest . \
\
\>", "Text",
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We could now say that it appears that (0, 1) is a local minimum and \
that (0, -1) is a saddle point .  Let's compare the heights of these two \
points:\
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This reinforces are finding, since the minimum is below the saddle \
point.  In fact, level curves around a local maximum or minimum will appear \
elliptic in shape and level curves around a saddle point appear hyperbolic in \
shape.  \t\
\>", "Text",
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Cell["\<\
Next, we pick a view point to help see the surface and the \
subsequent tangent plane defined from the linearization. This was obtained by \
trial and error.\
\>", "Text"],

Cell[BoxData[
    \(\(MView\  = \ {2.714, \ \(-1.902\), \ 1.448};\)\)], "Input"],

Cell["This is the actual surface as a 3D plot.", "Text"],

Cell[BoxData[
    \(\(actualsurfaceplot\  = \ 
        Plot3D[f[x, y], {x, \(-1\), 1}, {y, \(-1\), 1}, 
          ViewPoint \[Rule] MView];\)\)], "Input"],

Cell["\<\
The following complicated function generates the plot of a piece of \
the tangent plane at the designated point. Don't try to understand it yet; \
just evaluate it next.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(linearizedplot[a_, b_] := 
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          PlotPoints \[Rule] {2, 2}, 
          DisplayFunction \[Rule] Identity];\)\), "\n", 
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Cell["\<\
Now, we put everything together to get the surface, a piece of the \
tangent plane, and the point of tangency all in a single plot.\
\>", "Text"],

Cell[BoxData[
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        Axes \[Rule] Automatic, 
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Cell[TextData[{
  "The plot of ",
  Cell[BoxData[
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        FontWeight->"Bold"]]],
  " is that little bit of plane tangent to the surface at the plotted point \
",
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    FontWeight->"Bold"],
  ". "
}], "Text"],

Cell["Here is a plot of the actual error surface near (-.1,.4).", "Text"],

Cell[BoxData[
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Cell["See what happens for other some other points:", "Text"],

Cell[BoxData[{
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        AxesLabel \[Rule] {"\<x\>", "\<y\>", "\<f[x,y]\>"}, 
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Cell[CellGroupData[{

Cell["\<\
Lagrange's Method for Constrained Maximization and \
Minimization\
\>", "Section"],

Cell[TextData[StyleBox["Lagrange's method deals with maximizing or minimizing \
one function while another function is held constant. Once you are armed with \
the fact that the gradient gives you a normal vector, understanding it is \
pretty easy.",
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Cell["The Two Dimensional Case", "Subsection"],

Cell[TextData[{
  "If ",
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  " is on the level curve ",
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  " and ",
  Cell[BoxData[
      \(TraditionalForm\`\((x\_0, y\_0)\)\)]],
  " maximizes or minimizes f(x, y) given ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
  ", then how do you know that the level curve ",
  Cell[BoxData[
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  " is tangent to the level curve ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
  " at the point ",
  Cell[BoxData[
      \(TraditionalForm\`\((x\_0, y\_0)\)\)]],
  "?"
}], "Text"],

Cell[TextData[{
  " The first thing you can say is that ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y)\)]],
  " does not stay equal to  ",
  Cell[BoxData[
      \(TraditionalForm\`f(x\_0, y\_0)\)]],
  " on the level curve ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
  ". This means that you can leave ",
  Cell[BoxData[
      \(TraditionalForm\`\((x\_0, \ y\_0)\)\)]],
  " on the level curve ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
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  Cell[BoxData[
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  " and you can leave ",
  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  ".\nConsequently: If the level curve ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y)\  = \ f(x\_0, y\_0)\)]],
  " crosses ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
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  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
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  Cell[BoxData[
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  Cell[BoxData[
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  " given ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  " is tangent to the level curve ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
  " at the point ",
  Cell[BoxData[
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  "."
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Cell[CellGroupData[{

Cell[TextData[{
  "If ",
  Cell[BoxData[
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  " maximizes or minimizes ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y)\)]],
  " given ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
  ", you know that the level curve ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y)\  = \ f(x\_0, y\_0)\)]],
  " is tangent to the level curve ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
  " at the point ",
  Cell[BoxData[
      \(TraditionalForm\`\((x\_0, \ y\_0)\)\)]],
  ".\nWhy does this force\n     ",
  Cell[BoxData[
      \(gradf[x\_0, y\_0] = s\ gradg[x\_0, y\_0]\)]],
  "\nfor some number ",
  Cell[BoxData[
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  Cell[BoxData[
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  "? "
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  FontVariations->{"CompatibilityType"->0}],

Cell["Answer:", "Special1",
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Cell[TextData[{
  "\[Rule] The tangent to the level curve ",
  Cell[BoxData[
      \(f[x, y] = f[x\_0, y\_0]\)]],
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  Cell[BoxData[
      \(g[x, y] = c\)]],
  " at ",
  Cell[BoxData[
      \({x\_0, y\_0}\)]],
  ". \nSo any normal vector to the level curve \n       ",
  Cell[BoxData[
      \(f[x, y] = f[x\_0, y\_0]\)]],
  " at ",
  Cell[BoxData[
      \({x\_0, y\_0}\)]],
  " \nmust be parallel to any normal vector of \n       ",
  Cell[BoxData[
      \(g[x, y] = c\)]],
  " at ",
  Cell[BoxData[
      \({x\_0, y\_0}\)]],
  ".\n\[Rule] A normal vector to the level curve \n       ",
  Cell[BoxData[
      \(f[x, y] = f[x\_0, y\_0]\)]],
  " at ",
  Cell[BoxData[
      \({x\_0, y\_0}\)]],
  " is ",
  Cell[BoxData[
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  Cell[BoxData[
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  " at ",
  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  ".\nA compact way of saying this is to say\n     ",
  Cell[BoxData[
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  "\nfor some number ",
  Cell[BoxData[
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Cell[TextData[{
  "The numbers ",
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  " you get this way are called Lagrange multipliers by the heavy rollers."
}], "SmallText",
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Cell["\<\
Lagrange was a great French mathematician and astronomer 
who lived during the glory days 1736-1813.
\
\>", "Special2",
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  FontVariations->{"CompatibilityType"->0}]
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Cell[CellGroupData[{

Cell[TextData[{
  "What is Lagrange's method of maximizing or minimizing ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y)\)]],
  " given ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y)\  = \ c\)]],
  "?"
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  FontVariations->{"CompatibilityType"->0}],

Cell["Answer:", "Special1",
  FontFamily->"Times New Roman",
  FontVariations->{"CompatibilityType"->0}],

Cell[TextData[{
  "Lagrange's method of maximizing or minimizing \n",
  Cell[BoxData[
      \(f[x, y]\)]],
  " given ",
  Cell[BoxData[
      \(g[x, y] = c\)]],
  " is to solve the system of equations\n       ",
  Cell[BoxData[
      \(gradf[x, y] = s\ gradg[x, y]\)]],
  "\nand \n       ",
  Cell[BoxData[
      \(g[x, y] = c\)]],
  " \nfor ",
  Cell[BoxData[
      \({x, y}\)]],
  ".\nIf you get all the ",
  Cell[BoxData[
      \({x, y}\)]],
  "'s that solve this, then you are guaranteed that the true maximizers and \
minimizers are in this list.  "
}], "SmallText",
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  FontVariations->{"CompatibilityType"->0}]
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Cell[TextData[{
  "Look at the following movie:\nThe area of the bar graph on the right \
represents ",
  Cell[BoxData[
      \(1\/4\)]],
  " of the area of the inscribed rectangle."
}], "Text",
  FontFamily->"Times New Roman",
  FontVariations->{"CompatibilityType"->0}],

Cell[BoxData[{
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    \(\(rectangle[t_] = 
        Graphics[{Thickness[0.01], Blue, 
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          AxesOrigin \[Rule] {0, 0}, Ticks \[Rule] None, 
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Cell["Animate these and run at a slow speed.", "Special2"],

Cell[CellGroupData[{

Cell[TextData[{
  "For a given prescribed radius ",
  Cell[BoxData[
      \(TraditionalForm\`r\)]],
  ", use Lagrange's method to explain why the rectangle inscribed in the \
circle ",
  Cell[BoxData[
      \(TraditionalForm\`x\^2\  + \ y\^2\  = \ r\^2\)]],
  " with corners at ",
  Cell[BoxData[
      \(TraditionalForm\`\((x, \ 
        y)\), \ \((x, \ \(-y\))\), \ \((\(-x\), \ \(-y\))\)\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`\((\(-x\), \ y)\)\)]],
  " with the biggest possible area is a square."
}], "Text"],

Cell["Answer:", "Special1"],

Cell[TextData[{
  "Saying that ",
  Cell[BoxData[
      \({x, y}\)]],
  " is on the circle of radius ",
  Cell[BoxData[
      \(r\)]],
  " centered at ",
  Cell[BoxData[
      \({0, 0}\)]],
  " is to say that\n       ",
  Cell[BoxData[
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  ".\nFor a given ",
  Cell[BoxData[
      \({x, y}\)]],
  " on this circle, the area of the inscribed rectangle is\n       ",
  Cell[BoxData[
      \(f[x, y] = 4  x\ y\)]],
  ".\nTo find the largest such rectangle, you've got to maximize ",
  Cell[BoxData[
      \(f[x, y]\)]],
  " given ",
  Cell[BoxData[
      \(g[x, y] = r\^2\)]],
  ". \nThis is a snap for Lagrange's method:"
}], "SmallText"],

Cell[BoxData[{
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Cell[TextData[{
  "All four solutions for ",
  Cell[BoxData[
      \({x, y}\)]],
  " give a square; so the square is the largest inscribed rectangle.\nThis is \
not big news; you knew this all along.\nWhat might be good news is the \
method.  It was quick and easy."
}], "SmallText"]
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Cell["The Three Dimensional Case", "Subsection"],

Cell[CellGroupData[{

Cell[TextData[{
  "Does Lagrange's method work finding the maximizers of ",
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  " given ",
  Cell[BoxData[
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  "?"
}], "Text"],

Cell[TextData[{
  "In theory, yes.\nIf ",
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  Cell[BoxData[
      \(TraditionalForm\`g(x, y, z)\  = \ c\)]],
  " and the level surface ",
  Cell[BoxData[
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  " cuts ",
  Cell[BoxData[
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  " at ",
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  " and raise ",
  Cell[BoxData[
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  ". \nAnd you can leave ",
  Cell[BoxData[
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  " staying on the level surface ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y, z)\  = \ c\)]],
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  Cell[BoxData[
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  " and lower ",
  Cell[BoxData[
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  ". \nAs a result if ",
  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  "  must be tangent at ",
  Cell[BoxData[
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  ". \nSo, as in the two variable case, their normals must be parallel at ",
  Cell[BoxData[
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  ".\nThis means there is a number ",
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Cell[CellGroupData[{

Cell[TextData[{
  "Use Lagrange's method to find the maximum and minimum values (if any) of \
",
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      \(TraditionalForm\`f(x, y, z)\  = \ xyz\)]],
  "  given ",
  Cell[BoxData[
      \(TraditionalForm\`x\  + \ y\  + \ z\  = \ 6\)]],
  " with ",
  Cell[BoxData[
      \(TraditionalForm\`x\  > 0, \ y > 0, \ and\ \ z > 0\)]],
  "."
}], "Text"],

Cell[TextData[{
  "The set up is to maximize and minimize ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y, z)\  = \ xyz\)]],
  "  given ",
  Cell[BoxData[
      \(TraditionalForm\`x\  + \ y\  + \ z\  = \ 6\)]],
  " with ",
  Cell[BoxData[
      \(TraditionalForm\`x > 0, \ y > 0\ and\ z > 0\)]],
  ". You can push ",
  Cell[BoxData[
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  " close to (but not equal to) 0 while holding ",
  Cell[BoxData[
      \(TraditionalForm\`x\  + \ y\  + \ z\  = \ 6\)]],
  " by making ",
  Cell[BoxData[
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  " and ",
  Cell[BoxData[
      \(TraditionalForm\`y\)]],
  " small and positive and setting ",
  Cell[BoxData[
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  ". For ",
  Cell[BoxData[
      \(TraditionalForm\`x\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`y\)]],
  " small and positive,this makes ",
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  Cell[BoxData[
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  ":"
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    \(Clear[x, y, z, f, g]\), "\n", 
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Cell[TextData[{
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              StyleBox[ 
                AdjustmentBox[ 
                "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, 
                  BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", 
              AdjustmentBox[ 
              "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 
                0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", 
        "MMA"->"Mathematica"},
    Hyphenation->True,
    LineSpacing->{1, 3},
    ParagraphSpacing->{0, 6},
    CounterIncrements->"SmallText",
    FontFamily->"Helvetica",
    FontSize->9],
  
  Cell[StyleData["SmallText", "Presentation"],
    CellMargins->{{13, 10}, {8, 8}},
    LineSpacing->{1, 5},
    FontSize->12],
  
  Cell[StyleData["SmallText", "Condensed"],
    CellMargins->{{8, 10}, {2, 2}},
    LineSpacing->{1, 2},
    FontSize->9],
  
  Cell[StyleData["SmallText", "Printout"],
    CellMargins->{{9, 0}, {4, 4}},
    FontSize->7]
  }, Closed]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell["Styles for Input/Output", "Section"],
  
  Cell["\<\
The cells in this section define styles used for input and output \
to the kernel.  Be careful when modifying, renaming, or removing these \
styles, because the front end associates special meanings with these style \
names.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Input"],
    CellFrame->{{3, 0}, {0, 0}},
    CellMargins->{{52, 10}, {8, 8}},
    Evaluatable->True,
    CellGroupingRules->"InputGrouping",
    CellHorizontalScrolling->True,
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
    DefaultFormatType->DefaultInputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    LanguageCategory->"Formula",
    FormatType->InputForm,
    ShowStringCharacters->True,
    NumberMarks->True,
    LinebreakAdjustments->{0.85, 2, 10, 0, 1},
    CounterIncrements->"Input",
    FontWeight->"Bold",
    Background->RGBColor[1, 0.700008, 0.4]],
  
  Cell[StyleData["Input", "Presentation"],
    CellMargins->{{62, Inherited}, {10, 10}},
    LineSpacing->{1, 0}],
  
  Cell[StyleData["Input", "Condensed"],
    CellMargins->{{40, 10}, {4, 4}}],
  
  Cell[StyleData["Input", "Printout"],
    CellMargins->{{44, 0}, {6, 6}},
    LinebreakAdjustments->{0.85, 2, 10, 1, 1},
    Background->GrayLevel[0.8]]
  }, Closed]],
  
  Cell[StyleData["InlineInput"],
    Evaluatable->True,
    CellGroupingRules->"InputGrouping",
    CellHorizontalScrolling->True,
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    DefaultFormatType->DefaultInputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    AutoItalicWords->{},
    FormatType->InputForm,
    ShowStringCharacters->True,
    NumberMarks->True,
    CounterIncrements->"Input",
    FontWeight->"Bold"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Output"],
    CellFrame->{{3, 0}, {0, 0}},
    CellMargins->{{52, 10}, {8, 8}},
    CellEditDuplicate->True,
    CellGroupingRules->"OutputGrouping",
    CellHorizontalScrolling->True,
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    GeneratedCell->True,
    CellAutoOverwrite->True,
    CellLabelMargins->{{3, Inherited}, {Inherited, Inherited}},
    DefaultFormatType->DefaultOutputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    LanguageCategory->"Formula",
    FormatType->InputForm,
    CounterIncrements->"Output",
    Background->RGBColor[1, 0.900008, 0.900008]],
  
  Cell[StyleData["Output", "Presentation"],
    CellMargins->{{62, Inherited}, {12, 5}},
    LineSpacing->{1, 0}],
  
  Cell[StyleData["Output", "Condensed"],
    CellMargins->{{40, Inherited}, {4, 1}}],
  
  Cell[StyleData["Output", "Printout"],
    CellMargins->{{44, 0}, {6, 2}},
    Background->GrayLevel[0.900008]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Message"],
    CellMargins->{{62, Inherited}, {Inherited, Inherited}},
    CellGroupingRules->"OutputGrouping",
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    GeneratedCell->True,
    CellAutoOverwrite->True,
    ShowCellLabel->False,
    DefaultFormatType->DefaultOutputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    FormatType->InputForm,
    CounterIncrements->"Message",
    StyleMenuListing->None,
    FontColor->RGBColor[1, 0, 0]],
  
  Cell[StyleData["Message", "Presentation"],
    CellMargins->{{74, Inherited}, {Inherited, Inherited}},
    LineSpacing->{1, 0}],
  
  Cell[StyleData["Message", "Condensed"],
    CellMargins->{{50, Inherited}, {Inherited, Inherited}}],
  
  Cell[StyleData["Message", "Printout"],
    CellMargins->{{54, Inherited}, {Inherited, Inherited}},
    FontColor->GrayLevel[0]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Print"],
    CellMargins->{{62, Inherited}, {Inherited, Inherited}},
    CellGroupingRules->"OutputGrouping",
    CellHorizontalScrolling->True,
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    GeneratedCell->True,
    CellAutoOverwrite->True,
    ShowCellLabel->False,
    DefaultFormatType->DefaultOutputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    FormatType->InputForm,
    CounterIncrements->"Print",
    StyleMenuListing->None],
  
  Cell[StyleData["Print", "Presentation"],
    CellMargins->{{74, Inherited}, {Inherited, Inherited}},
    LineSpacing->{1, 0}],
  
  Cell[StyleData["Print", "Condensed"],
    CellMargins->{{50, Inherited}, {Inherited, Inherited}}],
  
  Cell[StyleData["Print", "Printout"],
    CellMargins->{{54, Inherited}, {Inherited, Inherited}}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Graphics"],
    CellMargins->{{62, Inherited}, {Inherited, Inherited}},
    CellGroupingRules->"GraphicsGrouping",
    CellHorizontalScrolling->True,
    PageBreakWithin->False,
    GeneratedCell->True,
    CellAutoOverwrite->True,
    ShowCellLabel->False,
    DefaultFormatType->DefaultOutputFormatType,
    FormatType->InputForm,
    CounterIncrements->"Graphics",
    StyleMenuListing->None],
  
  Cell[StyleData["Graphics", "Presentation"],
    CellMargins->{{74, Inherited}, {Inherited, Inherited}}],
  
  Cell[StyleData["Graphics", "Condensed"],
    CellMargins->{{52, Inherited}, {Inherited, Inherited}},
    ImageSize->{175, 175}],
  
  Cell[StyleData["Graphics", "Printout"],
    CellMargins->{{54, Inherited}, {Inherited, Inherited}},
    ImageSize->{250, 250}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["CellLabel"],
    StyleMenuListing->None,
    FontFamily->"Helvetica",
    FontSize->11,
    FontWeight->"Bold",
    FontColor->RGBColor[0.571389, 0.19675, 0.570504]],
  
  Cell[StyleData["CellLabel", "Presentation"],
    FontSize->12],
  
  Cell[StyleData["CellLabel", "Condensed"],
    FontSize->8],
  
  Cell[StyleData["CellLabel", "Printout"],
    FontSize->8,
    FontColor->GrayLevel[0]]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Unique Styles", "Section"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Author"],
    ShowCellBracket->False,
    CellMargins->{{10, 4}, {2, 10}},
    FontSize->16,
    FontSlant->"Italic"],
  
  Cell[StyleData["Author", "Presentation"],
    CellMargins->{{12, 10}, {2, 12}},
    LineSpacing->{1, 5},
    ParagraphSpacing->{0, 12},
    FontSize->20],
  
  Cell[StyleData["Author", "Condensed"],
    CellMargins->{{8, 10}, {1, 4}},
    LineSpacing->{1, 1},
    ParagraphSpacing->{0, 4},
    FontSize->12],
  
  Cell[StyleData["Author", "Printout"],
    CellMargins->{{9, 0}, {4, 12}},
    ParagraphSpacing->{0, 6},
    FontSize->14]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Copyright"],
    ShowCellBracket->False,
    CellMargins->{{10, 10}, {40, 2}},
    FontFamily->"Helvetica",
    FontSize->9],
  
  Cell[StyleData["Copyright", "Presentation"],
    CellMargins->{{12, 10}, {50, 2}},
    LineSpacing->{1, 5},
    FontSize->12],
  
  Cell[StyleData["Copyright", "Condensed"],
    CellMargins->{{8, 10}, {12, 1}},
    LineSpacing->{1, 2},
    FontSize->9],
  
  Cell[StyleData["Copyright", "Printout"],
    CellMargins->{{9, 0}, {72, 4}},
    FontSize->7]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Example"],
    CellMargins->{{12, 10}, {5, 12}},
    LineSpacing->{1, 3},
    ParagraphSpacing->{0, 12},
    CounterIncrements->"Example",
    FontFamily->"Times",
    FontWeight->"Bold"],
  
  Cell[StyleData["Example", "Presentation"],
    CellMargins->{{18, 10}, {8, 20}},
    LineSpacing->{1, 5},
    ParagraphSpacing->{0, 12}],
  
  Cell[StyleData["Example", "Condensed"],
    CellMargins->{{8, 10}, {4, 8}},
    LineSpacing->{1, 1},
    ParagraphSpacing->{0, 4}],
  
  Cell[StyleData["Example", "Printout"],
    CellMargins->{{9, 0}, {4, 10}},
    ParagraphSpacing->{0, 6}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Definition"],
    CellFrame->{{3, 0}, {0, 0}},
    CellMargins->{{52, 10}, {8, 8}},
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}},
    Hyphenation->True,
    ShowStringCharacters->True,
    CounterIncrements->"Definition",
    FontFamily->"Helvetica",
    FontWeight->"Bold",
    FontColor->GrayLevel[1],
    Background->RGBColor[0.2, 0.700008, 0.700008]],
  
  Cell[StyleData["Definition", "Presentation"],
    CellMargins->{{62, Inherited}, {5, 12}},
    LineSpacing->{1, 0}],
  
  Cell[StyleData["Definition", "Condensed"],
    CellMargins->{{40, 10}, {1, 4}}],
  
  Cell[StyleData["Definition", "Printout"],
    CellMargins->{{44, 0}, {2, 6}},
    Background->GrayLevel[0.6]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Theorem"],
    CellFrame->{{3, 0}, {0, 0}},
    CellMargins->{{52, 10}, {8, 8}},
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}},
    Hyphenation->True,
    ShowStringCharacters->True,
    CounterIncrements->"Theorem",
    FontFamily->"Helvetica",
    FontWeight->"Bold",
    FontColor->GrayLevel[1],
    Background->RGBColor[0.571389, 0.19675, 0.570504]],
  
  Cell[StyleData["Theorem", "Presentation"],
    CellMargins->{{62, Inherited}, {5, 12}},
    LineSpacing->{1, 0}],
  
  Cell[StyleData["Theorem", "Condensed"],
    CellMargins->{{40, 10}, {1, 4}}],
  
  Cell[StyleData["Theorem", "Printout"],
    CellMargins->{{44, 0}, {2, 6}},
    Background->GrayLevel[0.4]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["ExerciseMain"],
    CellFrame->{{6, 0}, {0, 1}},
    CellMargins->{{12, Inherited}, {4, 20}},
    CellGroupingRules->{"SectionGrouping", 30},
    PageBreakBelow->False,
    CellFrameColor->RGBColor[0.571389, 0.19675, 0.570504],
    CounterIncrements->"Subsection",
    CounterAssignments->{{"Subsubsection", 0}},
    FontFamily->"Helvetica",
    FontSize->16,
    FontWeight->"Bold",
    FontColor->RGBColor[0.571389, 0.19675, 0.570504]],
  
  Cell[StyleData["ExerciseMain", "Presentation"],
    CellMargins->{{18, 10}, {8, 32}},
    LineSpacing->{1, 2},
    FontSize->24,
    FontTracking->"Condensed"],
  
  Cell[StyleData["ExerciseMain", "Condensed"],
    CellMargins->{{8, Inherited}, {2, 12}},
    FontSize->12],
  
  Cell[StyleData["ExerciseMain", "Printout"],
    CellMargins->{{9, 0}, {2, 50}},
    CellFrameColor->GrayLevel[0.500008],
    FontSize->14,
    FontTracking->"Plain",
    FontColor->GrayLevel[0]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Exercise"],
    CellDingbat->"\[FilledDownTriangle]",
    CellMargins->{{23, Inherited}, {4, 18}},
    CellGroupingRules->{"SectionGrouping", 50},
    PageBreakBelow->False,
    Hyphenation->True,
    CounterIncrements->"Subsubsection",
    FontFamily->"Times",
    FontSize->13,
    FontWeight->"Bold",
    FontColor->RGBColor[0.571389, 0.19675, 0.570504]],
  
  Cell[StyleData["Exercise", "Presentation"],
    CellMargins->{{33, 10}, {8, 26}},
    LineSpacing->{1, 0},
    FontSize->18],
  
  Cell[StyleData["Exercise", "Condensed"],
    CellMargins->{{17, Inherited}, {2, 12}},
    FontSize->10],
  
  Cell[StyleData["Exercise", "Printout"],
    CellFrame->{{0, 0}, {0.5, 0}},
    CellDingbat->None,
    CellMargins->{{9, 0}, {6, 20}},
    FontSize->11,
    FontColor->GrayLevel[0]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["ExerciseText"],
    CellMargins->{{62, 10}, {5, 5}},
    Hyphenation->True,
    LineSpacing->{1, 3},
    ParagraphSpacing->{0, 8},
    CounterIncrements->"ExcersiceText",
    FontFamily->"Times"],
  
  Cell[StyleData["ExerciseText", "Presentation"],
    CellMargins->{{74, 10}, {8, 8}},
    LineSpacing->{1, 5},
    ParagraphSpacing->{0, 12}],
  
  Cell[StyleData["ExerciseText", "Condensed"],
    CellMargins->{{52, 10}, {2, 2}},
    LineSpacing->{1, 1},
    ParagraphSpacing->{0, 4}],
  
  Cell[StyleData["ExerciseText", "Printout"],
    CellMargins->{{54, 0}, {4, 4}},
    ParagraphSpacing->{0, 6}]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Formulas and Programming", "Section"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["DisplayFormula"],
    CellMargins->{{62, 10}, {2, 10}},
    CellHorizontalScrolling->True,
    DefaultFormatType->DefaultInputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    LanguageCategory->"Formula",
    ScriptLevel->0,
    SingleLetterItalics->True,
    UnderoverscriptBoxOptions->{LimitsPositioning->True}],
  
  Cell[StyleData["DisplayFormula", "Presentation"],
    CellMargins->{{74, 10}, {2, 10}},
    FontSize->10],
  
  Cell[StyleData["DisplayFormula", "Condensed"],
    CellMargins->{{52, 10}, {2, 10}},
    FontSize->10],
  
  Cell[StyleData["DisplayFormula", "Printout"],
    CellMargins->{{54, 10}, {2, 10}},
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["ChemicalFormula"],
    CellMargins->{{62, 10}, {2, 10}},
    DefaultFormatType->DefaultInputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    LanguageCategory->"Formula",
    AutoSpacing->False,
    ScriptLevel->1,
    ScriptBaselineShifts->{0.6, Automatic},
    SingleLetterItalics->False,
    ZeroWidthTimes->True],
  
  Cell[StyleData["ChemicalFormula", "Presentation"],
    CellMargins->{{74, 10}, {2, 10}},
    FontSize->10],
  
  Cell[StyleData["ChemicalFormula", "Condensed"],
    CellMargins->{{52, 10}, {2, 10}},
    FontSize->10],
  
  Cell[StyleData["ChemicalFormula", "Printout"],
    CellMargins->{{54, 10}, {2, 10}},
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Program"],
    CellMargins->{{12, 10}, {Inherited, 6}},
    Hyphenation->False,
    LanguageCategory->"Formula",
    FontFamily->"Courier"],
  
  Cell[StyleData["Program", "Presentation"],
    CellMargins->{{13, 30}, {Inherited, 4}},
    FontSize->9.5],
  
  Cell[StyleData["Program", "Condensed"],
    CellMargins->{{8, 10}, {Inherited, 4}},
    FontSize->9.5],
  
  Cell[StyleData["Program", "Printout"],
    CellMargins->{{9, 0}, {Inherited, 4}},
    FontSize->9.5]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Styles for Automatic Numbering", "Section"],
  
  Cell["\<\
The following styles are useful for numbered equations, figures, \
etc.  They automatically give the cell a FrameLabel containing a reference to \
a particular counter, and also increment that counter.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["NumberedEquation"],
    CellMargins->{{62, 10}, {Inherited, Inherited}},
    CellFrameLabels->{{None, Cell[ 
            TextData[ {"(", 
              CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}},
    DefaultFormatType->DefaultInputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    CounterIncrements->"NumberedEquation",
    FormatTypeAutoConvert->False],
  
  Cell[StyleData["NumberedEquation", "Presentation"],
    CellMargins->{{74, 10}, {Inherited, Inherited}}],
  
  Cell[StyleData["NumberedEquation", "Condensed"],
    CellMargins->{{52, 10}, {Inherited, Inherited}}],
  
  Cell[StyleData["NumberedEquation", "Printout"],
    CellMargins->{{54, 0}, {Inherited, Inherited}}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["NumberedFigure"],
    CellMargins->{{62, 145}, {Inherited, Inherited}},
    CellFrameLabels->{{None, None}, {Cell[ 
            TextData[ {"Figure ", 
              CounterBox[ "NumberedFigure"]}]], None}},
    CounterIncrements->"NumberedFigure",
    ImageMargins->{{43, Inherited}, {Inherited, 0}},
    FormatTypeAutoConvert->False],
  
  Cell[StyleData["NumberedFigure", "Presentation"]],
  
  Cell[StyleData["NumberedFigure", "Condensed"]],
  
  Cell[StyleData["NumberedFigure", "Printout"]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["NumberedTable"],
    CellMargins->{{62, 145}, {Inherited, Inherited}},
    CellFrameLabels->{{None, None}, {Cell[ 
            TextData[ {"Table ", 
              CounterBox[ "NumberedTable"]}]], None}},
    TextAlignment->Center,
    CounterIncrements->"NumberedTable",
    FormatTypeAutoConvert->False],
  
  Cell[StyleData["NumberedTable", "Presentation"]],
  
  Cell[StyleData["NumberedTable", "Condensed"]],
  
  Cell[StyleData["NumberedTable", "Printout"]]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Styles for Headers and Footers", "Section"],
  
  Cell[StyleData["Header"],
    CellMargins->{{0, 0}, {4, 1}},
    StyleMenuListing->None,
    FontFamily->"Helvetica",
    FontSize->9,
    FontSlant->"Italic"],
  
  Cell[StyleData["Footer"],
    CellMargins->{{0, 0}, {0, 4}},
    StyleMenuListing->None,
    FontFamily->"Helvetica",
    FontSize->6],
  
  Cell[StyleData["PageNumber"],
    CellMargins->{{0, 0}, {4, 1}},
    StyleMenuListing->None,
    FontFamily->"Helvetica",
    FontSize->9,
    FontWeight->"Bold"]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Hyperlink Styles", "Section"],
  
  Cell["\<\
The cells below define styles useful for making hypertext \
ButtonBoxes.  The \"Hyperlink\" style is for links within the same Notebook, \
or between Notebooks.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Hyperlink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->GrayLevel[1],
    Background->RGBColor[1, 0.4, 0],
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`NotebookLocate[ #2]}]&),
    Active->True,
    ButtonFrame->"None",
    ButtonNote->ButtonData}],
  
  Cell[StyleData["Hyperlink", "Presentation"]],
  
  Cell[StyleData["Hyperlink", "Condensed"]],
  
  Cell[StyleData["Hyperlink", "Printout"],
    FontColor->GrayLevel[0],
    Background->GrayLevel[1]]
  }, Closed]],
  
  Cell["\<\
The following styles are for linking automatically to the on-line \
help system.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["MainBookLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->GrayLevel[1],
    Background->RGBColor[1, 0.4, 0],
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["MainBookLink", "Presentation"]],
  
  Cell[StyleData["MainBookLink", "Condensed"]],
  
  Cell[StyleData["MainBookLink", "Printout"],
    FontColor->GrayLevel[0],
    Background->GrayLevel[1]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["AddOnsLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontFamily->"Courier",
    FontColor->GrayLevel[1],
    Background->RGBColor[1, 0.4, 0],
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["AddOnsLink", "Presentation"]],
  
  Cell[StyleData["AddOnsLink", "Condensed"]],
  
  Cell[StyleData["AddOnLink", "Printout"],
    FontColor->GrayLevel[0],
    Background->GrayLevel[1]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["RefGuideLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontFamily->"Courier",
    FontColor->GrayLevel[1],
    Background->RGBColor[1, 0.4, 0],
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["RefGuideLink", "Presentation"]],
  
  Cell[StyleData["RefGuideLink", "Condensed"]],
  
  Cell[StyleData["RefGuideLink", "Printout"],
    FontColor->GrayLevel[0],
    Background->GrayLevel[1]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["GettingStartedLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->GrayLevel[1],
    Background->RGBColor[1, 0.4, 0],
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["GettingStartedLink", "Presentation"]],
  
  Cell[StyleData["GettingStartedLink", "Condensed"]],
  
  Cell[StyleData["GettingStartedLink", "Printout"],
    FontColor->GrayLevel[0],
    Background->GrayLevel[1]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["OtherInformationLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->GrayLevel[1],
    Background->RGBColor[1, 0.4, 0],
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["OtherInformationLink", "Presentation"]],
  
  Cell[StyleData["OtherInformationLink", "Condensed"]],
  
  Cell[StyleData["OtherInformationLink", "Printout"],
    FontColor->GrayLevel[0],
    Background->GrayLevel[1]]
  }, Closed]]
  }, Closed]],
  
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